Trisecting an Angle Using Tschirnhaus's Cubic

This Demonstration illustrates a property of Tschirnhaus's cubic, which has polar equation . Namely, that the angle between the tangent and the normal to the radius vector at a given point on the curve is one-third of the polar angle of the point.

To trisect a given angle , draw the radius vector (red) from the origin, making that angle with the axis, to meet at a point on the curve. Construct the tangent (green) and the normal to the radius vector (green) at the point. The angle between these two lines is . So the angle is .